geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the polyhedral groups are the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s of the

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

Groups

There are three polyhedral groups: *The

tetrahedral group 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...

of order 12, rotational symmetry group of the

regular tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

. It is isomorphic to A₄. ** The

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...

es of T are: *** identity *** 4 × rotation by 120°, order 3, cw *** 4 × rotation by 120°, order 3, ccw *** 3 × rotation by 180°, order 2 *The

octahedral group A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

of order 24, rotational symmetry group of the

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

and the

regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

. It is isomorphic to S₄. ** The conjugacy classes of O are: *** identity *** 6 × rotation by ±90° around vertices, order 4 *** 8 × rotation by ±120° around triangle centers, order 3 *** 3 × rotation by 180° around vertices, order 2 *** 6 × rotation by 180° around midpoints of edges, order 2 * The

icosahedral group In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of th ...

of order 60, rotational symmetry group of the

regular dodecahedron A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...

and the

regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...

. It is isomorphic to A₅. ** The conjugacy classes of I are: *** identity *** 12 × rotation by ±72°, order 5 *** 12 × rotation by ±144°, order 5 *** 20 × rotation by ±120°, order 3 *** 15 × rotation by 180°, order 2 These symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, ,3can be seen as the union of 6 tetrahedral symmetry ,3mirrors, and 3 mirrors of

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

Dih₂, ,2

Pyritohedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...

is another doubling of tetrahedral symmetry. The conjugacy classes of full tetrahedral symmetry, , are: * identity * 8 × rotation by 120° * 3 × rotation by 180° * 6 × reflection in a plane through two rotation axes * 6 × rotoreflection by 90° The conjugacy classes of pyritohedral symmetry, T_h, include those of T, with the two classes of 4 combined, and each with inversion: * identity * 8 × rotation by 120° * 3 × rotation by 180° * inversion * 8 × rotoreflection by 60° * 3 × reflection in a plane The conjugacy classes of the full octahedral group, , are: * inversion * 6 × rotoreflection by 90° * 8 × rotoreflection by 60° * 3 × reflection in a plane perpendicular to a 4-fold axis * 6 × reflection in a plane perpendicular to a 2-fold axis The conjugacy classes of full icosahedral symmetry, , include also each with inversion: * inversion * 12 × rotoreflection by 108°, order 10 * 12 × rotoreflection by 36°, order 10 * 20 × rotoreflection by 60°, order 6 * 15 × reflection, order 2

Chiral polyhedral groups

Full polyhedral groups

References

* Coxeter, H. S. M.

Regular Polytopes ''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...

, 3rd ed. New York: Dover, 1973. (''The Polyhedral Groups''. §3.5, pp. 46–47)

External links

* {{mathworld , urlname = PolyhedralGroup, title =PolyhedralGroup Polyhedra

Groups

Chiral polyhedral groups

Full polyhedral groups

See also

References

External links